Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Wednesday, July 27, 2005

Majority is Stablest

Consider the following two voting schemes to elect a single candidate.

Majority Vote.

A Majority of Majorities (think an electoral college system with
states of equal size).

Which of these voting systems are more stable, i.e., less likely to be
affected by flipping a small number of votes?

In an upcoming FOCS paper, Elchanan
Mossel, Ryan O'Donnell and Krzysztof Oleszkiewicz prove the
"Majority is Stablest" conjecture that answers the above
question and in fact shows that majority is the most stable function
among balanced Boolean functions where each input has low
influence. To understand this result we'll need to define the terms in
the statement of the theorem.

Balanced: A Boolean function is balanced if it has the same number
of inputs mapping to zero as mapping to one.

The influence of the ith variable is the expectation over a
random input of the variance of setting the ith bit of the
input randomly. The conjecture requires the influence of each variable
to be bounded by a small constant.

Stability: The noise stability of f is the expectation of f(x)f(y)
where x and y are chosen independently.

I keep seeing results about one-time voting systems, including instant-runoff voting or approval voting, but how about results where parties get to vote in multiple elections? (For example, real runoff elections.)